Some curvature properties of trans-Sasakian manifolds

c Pleiades Publishing, Ltd., 2014. ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2014, Vol. 35, No. 2, pp. 56–64.  Some Curvature Properties of Trans-Sasakian Manifolds Ali Akbar* and Avijit Sarkar** (Submitted by M. A. Malakhaltsev) Department of Mathematics, University of Kalyani, Kalyani-741235, West Bengal, India Received August 23, 2013 Abstract—The object of the present paper is to study quasi-conformally flat trans-Sasakian manifolds. We consider trans-Sasakian manifolds with η-parallel and cyclic parallel Ricci tensors. φ-Ricci symmetric quasi-conformally flat trans-Sasakian manifolds have been studied. We also investigates quasi-conformally flat trans-Sasakian manifolds which are Einstein Semi-symmetric. DOI: 10.1134/S1995080214020024 Keywords and phrases: trans-Sasakian manifolds, locally φ-symmetric, η-parallel Ricci tensor, quasi-conformal curvature tensor and Einstein Semisymmetric. 1. INTRODUCTION In 1985 J.A. Oubina [7] introduced a new class of almost contact metric manifolds, called transSasakian manifolds, which includes Sasakian, Kenmotso and Cosymplectic structures. The authors in the paper [1, 3] and [12] studied such manifolds and obtained some interesting results. In the paper [5] the author studied conformally flat φ-recurrent trans-Sasakian manifolds. It is known that [6] transSasakian structure of type (0, 0), (0, β) and (α, 0) are Cosymplectic, β-Kenmotsu and α-Sasakian respectively, where α, β ∈ R. In [8] J.C. Marrero proved that a trans-Sasakian manifold of dimension n ≥ 5 is either a Cosymplectic manifold or an α-Sasakian or a β-Kenmotsu manifold. In the present paper I have studied quasi-conformally flat trans-Sasakian manifolds of dimension n ≥ 5. Such manifolds are either a cosymplectic or a β-Kenmotsu or an α-Sasakian. The present paper is organized as follows: After introduction in Section 1, some preliminaries are given in Section 2. In Section 3 I study quasiconformally flat trans-Sasakian manifolds. Section 4 is devoted to study trans-Sasakian manifolds with η-parallel and cyclic parallel Ricci tensors. φ-Ricci symmetric quasi-conformally flat trans-Sasakian manifolds are also considered in this section. In section 5 I study quasi-conformally flat trans-Sasakian manifolds which are Einstein Semi-symmetric. 2. PRELIMINARIES Let M be a (2n + 1)-dimensional connected differentiable manifold endowed with an almost contact metric structure (φ, ξ, η, g), where φ is a tensor field of type (1, 1), ξ is a vector field, η is an 1-form and g is a Riemannian metric on M such that [2] φ2 X = −X + η(X)ξ, η(ξ) = 1. g(φX, φY ) = g(X, Y ) − η(X)η(Y ). X, Y ∈ T (M ) (2.1) (2.2) Then also φξ = 0, η(φX) = 0, η(X) = g(X, ξ). g(φX, X) = 0. * ** E-mail: [email protected] E-mail: [email protected] 56 (2.3) (2.4) SOME CURVATURE PROPERTIES 57 An almost contact metric manifold M 2n+1 (φ, ξ, η, g) is said to be a trans-Sasakian manifold [7] if (M 2n+1 × R, J, G) belongs to the class W4 [4] of the Hermitian manifolds, where J is the almost complex structure on M 2n+1 × R defined by [13]     d d J Z, f = φZ − f ξ, η(Z) , (2.5) dt dt for any vector field Z on M 2n+1 and smooth function f on M 2n+1 × R and G is the Hermitian metric on the product M 2n+1 × R. This may be expressed by the condition [7] (∇X φ)Y = α(g(X, Y )ξ − η(Y )X) + β(g(φX, Y )ξ − η(Y )φX), (2.6) for some smooth functions α and β on M 2n+1 , and we say that the trans-Sasakian structure is of type (α, β) . From equation (2.6), it follows that ∇X ξ = −αφX + β(X − η(Y )ξ), (2.7) (∇X η)Y = −αg(φX, Y )ξ + βg(φX, φY ). (2.8) In a (2n + 1)-dimensional trans-Sasakian manifold, from (2.6), (2.7) and (2.8), we can write [12] R(X, Y )ξ = (α2 − β 2 ){η(Y )X − η(X)Y } + 2αβ{η(Y )φX − η(X)φY } − (Xα)φY + (Y α)φX − (Xβ)φ2 Y + (Y β)φ2 X. (2.9) S(X, ξ) = {2n(α2 − β 2 ) − ξβ}η(X) − (2n − 1)Xβ − (φX)α, (2.10) where S is the Ricci tensor. Further, we have 2αβ + ξα = 0. (2.11) 3. QUASI-CONFORMALLY FLAT TRANS-SASAKIAN MANIFOLD Definition 3.1. The notion of quasi-conformal curvature tensor was given by Yano and Sawaki [15]. According to them a quasi-conformal curvature tensor C is given by C(X, Y )Z = aR(X, Y )Z + b{S(Y, Z)X − S(X, Z)Y + g(Y, Z)QX − g(X, Z)QY }  r a + 2b {g(Y, Z)X − g(X, Z)Y }, (3.1) − 2n + 1 2b where a, b are constants R, Q and r are the Reimanian curvature tensor of type (1, 3), the Ricci operator defined by g(QX, Y ) = S(X, Y ) and the scalar curvature, respectively. Definition 3.2. A (2n + 1)-dimensional trans-Sasakian manifold will be called quasi-conformally flat if C(X, Y )Z = 0, for any vector fields X, Y , Z. Definition 3.3. A (2n + 1)-dimensional quasi-conformally flat trans-Sasakian manifold will be called locally φ-symmetric if, φ2 (∇W R)(X, Y )Z = 0, for any vector fields X, Y , Z and W . In this connection it should be mentioned that the notion of locally φ- symmetric manifolds was introduced by T. Takahashi [14] in the context of Sasakian geometry. Definition 3.4. A (2n + 1)-dimensional quasi-conformally flat trans-Sasakian manifold will be called φ-recurrent if, φ2 (∇W R)(X, Y )Z = A(W )R(X, Y )Z for any vector fields X, Y , Z and W . In this connection it should be mentioned that the notion of locally φ-recurrent manifolds was introduced in the paper [9] in context of Sasakian geometry. Definition 3.5. If an almost contact Riemannian manifold M satisfies the condition S = ag + bη ⊗ η (3.2) for some functions a, b in C ∞ (M ) and S is the Ricci tensor, then M is said to be an η-Einstein manifold. If, in particular, a = 0 then this manifold will be called a special type of η-Einstein manifold. Now, we consider a trans-Sasakian manifold which is quasi-conformally flat. Then we get from (3.1) b R(X, Y )Z = − {S(Y, Z)X − S(X, Z)Y + g(Y, Z)QX − g(X, Z)QY } a LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 35 No. 2 2014 58 ALI AKBAR, AVIJIT SARKAR + a  r + 2b {g(Y, Z)X − g(X, Z)Y }. a(2n + 1) 2b (3.3) Taking inner product on both side of above with respect to W , we get b R(X, Y, Z, W ) = − {S(Y, Z)g(X, W ) − S(X, Z)g(Y, W ) + g(Y, Z)g(QX, W ) a a  r + 2b {g(Y, Z)g(X, W ) − g(X, Z)g(Y, W )} − g(X, Z)g(QY, W )} + a(2n + 1) 2b (3.4) Where, R(X, Y, Z, W ) = g(R(X, Y )Z, W ). Now using g(QX, Y ) = S(X, Y ) in above we get, b R(X, Y, Z, W ) = − {S(Y, Z)g(X, W ) − S(X, Z)g(Y, W ) + g(Y, Z)S(X, W ) a a  r − g(X, Z)S(Y, W )} + + 2b {g(Y, Z)g(X, W ) − g(X, Z)g(Y, W )}. a(2n + 1) 2b (3.5) Again from (2.9) we get R(ξ, X, Y, ξ) = (α2 − β 2 − ξβ)g(φX, φY ). (3.6) Suppose α and β are constant. Then we get from (2.10), (3.6) S(X, ξ) = 2n(α2 − β 2 )η(X). (3.7) S(ξ, ξ) = 2n(α2 − β 2 ). (3.8) R(ξ, X, Y, ξ) = (α2 − β 2 )g(φX, φY ). (3.9) Putting X = W = ξ in equation (3.5) we get b R(ξ, Y, Z, ξ) = − {S(Y, Z) − S(ξ, Z)η(Y ) + g(Y, Z)S(ξ, ξ) − η(Z)S(Y, ξ)} a a  r + 2b {g(Y, Z) − η(Z)η(Y )}. + a(2n + 1) 2b (3.10) Using (3.7), (3.8) and (3.9) in (3.10) we get b (α2 − β 2 )g(φY, φZ) = − {S(Y, Z) − 2n(α2 − β 2 )η(Z)η(Y ) + 2n(α2 − β 2 )g(Y, Z) a a  r 2 2 + 2b {g(Y, Z) − η(Z)η(Y )}, − 2n(α − β )η(Y )η(Z)} + a(2n + 1) 2b (3.11) or  a   a 2 r 2 (α − β ) g(Y, Z) + 2b + 2n − S(Y, Z) = b(2n + 1) 2b b   a   r a 2 + − (α − β 2 ) η(Y )η(Z), + 2b + 4n + b(2n + 1) 2b b  (3.12) or S(Y, Z) = Ag(Y, Z) + Bη(Y )η(Z), (3.13) where A=   a   r a 2 2 (α − β ) , + 2b + 2n − b(2n + 1) 2b b (3.14) and  B= −  a   r a 2 2 (α − β ) . + 2b + 4n + b(2n + 1) 2b b (3.15) LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 35 No. 2 2014 SOME CURVATURE PROPERTIES 59 Using g(QX, Y ) = S(X, Y ) in (3.13) we get, QX = AX + Bη(X)ξ. (3.16) Thus, we are in a position to state the following: Theorem 3.1. A quasi-conformally flat trans-Sasakian manifolds of dimension n ≥ 5 is η-Einstein. Again by [8] a trans-Sasakian manifold of dimension n ≥ 5 is either a Cosymplectic manifold or an α-Sasakian manifold or a β-Kenmotsu manifold. Thus we have the following corollaries: Corollary 3.1. A quasi-conformally flat Cosymplectic manifold of dimension n ≥ 5 is η-Einstein. Corollary 3.2. A quasi-conformally flat α-Sasakian manifold of dimension n ≥ 5 is η-Einstein. Corollary 3.3. A quasi-conformally flat β-Kenmotsu manifold of dimension n ≥ 5 is η-Einstein. Again using (3.12) and (3.16) in (3.3) we get   a  2b  r a 2 2 R(X, Y )Z = − 2n − (α − β ) {g(Y, Z)X + 2b − a(2n + 1) 2b a b   a  b a 2 r 2 − g(X, Z)Y } + 4n + (α − β ) {g(Y, Z)η(X)ξ + 2b − a(2n + 1) 2b a b − g(X, Z)η(Y )ξ + η(Y )η(Z)X − η(X)η(Z)Y }. (3.17) Now differentiating covariantly with respect to W we get from (3.17)  dr(W )  a + 2b {g(Y, Z)X − g(X, Z)Y } (∇W R)(X, Y )Z = − a(2n + 1) 2b    dr(W ) a + + 2b {g(Y, Z)η(X)ξ − g(X, Z)η(Y )ξ + η(Y )η(Z)X − η(X)η(Z)Y } a(2n + 1) 2b   a  b r a 2 2 4n + (α − β ) {g(Y, Z)η(X)(∇W ξ) + + 2b − a(2n + 1) 2b a b + g(Y, Z)(∇W η)(X)ξ − g(X, Z)η(Y )(∇W ξ) − g(X, Z)(∇W η)(Y )ξ + (∇W η)(Y )η(Z)X + (∇W η)(Z)η(Y )X − (∇W η)(X)η(Z)Y − (∇W η)(Z)η(X)Y }. (3.18) Taking X, Y , Z and W orthogonal to ξ and applying φ2 on both side of above we get,  dr(W )  a φ2 (∇W R)(X, Y )Z = + 2b {g(Y, Z)X − g(X, Z)Y }. a(2n + 1) 2b (3.19) Thus we are in a position to state the following: Theorem 3.2. A quasi-conformally flat trans-Sasakian manifold of dimension n ≥ 5 is locally φsymmetric if and only if the scalar curvature is constant provided that α and β are constants. Again by [8] a trans-Sasakian manifold of dimension n ≥ 5 is either a Cosymplectic manifold or an α-Sasakian manifold or a β-Kenmotsu manifold. Thus we have the following corollaries: Corollary 3.4. A quasi-conformally flat Cosymplectic manifold of dimension n ≥ 5 is locally φsymmetric if and only if the scalar curvature is constant provided that α and β are constants. Corollary 3.5. A quasi-conformally flat α-Sasakian manifold of dimension n ≥ 5 is locally φsymmetric if and only if the scalar curvature is constant provided that α and β are constants. Corollary 3.6. A quasi-conformally flat β-Kenmotsu manifold of dimension n ≥ 5 is locally φsymmetric if and only if the scalar curvature is constant provided that α and β are constants. Remark 3.1. In this connection it should be mentioned that in paper [10] the authors have proved that a three dimensional trans-Sasakian manifold is locally φ-symmetric if and only if the scalar curvature is constant provided that α and β are constants. Again we suppose that the quasi-conformally flat trans-Sasakian manifold is φ-recurrent. Then we have φ2 (∇W R)(X, Y )Z = A(W )R(X, Y )Z. (3.20) LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 35 No. 2 2014 60 ALI AKBAR, AVIJIT SARKAR Using (3.19) we get from above,  dr(W )  a + 2b {g(Y, Z)X − g(X, Z)Y }. a(2n + 1) 2b (3.21)  dr(W )  a 1 + 2b {g(Y, Z)X − g(X, Z)Y }. A(W ) a(2n + 1) 2b (3.22) A(W )R(X, Y )Z = or, R(X, Y )Z = Putting W = ei in the above equation, where {ei } is an orthonormal basis of the tangent space at any point of the manifold and taking summation over i, 1 ≤ i ≤ 2n + 1, we get  1 dr(ei )  a + 2b {g(Y, Z)X − g(X, Z)Y }. (3.23) R(X, Y )Z = A(ei ) a(2n + 1) 2b or, R(X, Y )Z = λ{g(Y, Z)X − g(X, Z)Y } (3.24) dr(ei ) 1 where λ = A(e ( a + 2b) is a scalar. Since A is non zero, λ will be constant. Therefore M is of i ) a(2n+1) 2b constant curvature λ. Thus we are in a position to state the following: Theorem 3.3. A φ-recurrent quasi-conformally flat trans-Sasakian manifold of dimension n ≥ 5 is a manifold of constant curvature, provided that α and β are constants. Again by [8] a trans-Sasakian manifold of dimension n ≥ 5 is either a Cosymplectic manifold or an α-Sasakian manifold or a β-Kenmotsu manifold. Thus we have the following corollaries: Corollary 3.7. A φ-recurrent quasi-conformally flat Cosymplectic manifold of dimension n ≥ 5 is a manifold of constant curvature, provided that α and β are constants. Corollary 3.8. A φ-recurrent quasi-conformally flat α-Sasakian manifold of dimension n ≥ 5 is a manifold of constant curvature, provided that α and β are constants. Corollary 3.9. A φ-recurrent quasi-conformally flat β-Kenmotsu manifold of dimension n ≥ 5 is a manifold of constant curvature, provided that α and β are constants. 4. η-PARALLEL, CYCLIC PARALLEL RICCI TENSORS AND φ-RICCI SYMMETRIC QUASI-CONFORMALLY FLAT TRANS-SASAKIAN MANIFOLD Definition 4.1. The Ricci tensor S of a quasi-conformally flat trans-Sasakian manifold will be called η-parallel if it satisfies (∇X S)(φY, φZ)= 0 , for any vector fields X, Y , Z. From (3.12) we get,       dr(W ) dr(W ) a a + 2b g(Y, Z) + − + 2b η(Y )η(Z) (∇X S)(Y, Z) = b(2n + 1) 2b b(2n + 1) 2b       r a a + − + 2b + 4n + (α2 − β 2 ) {(∇X η)(Y )η(Z) + η(Y )(∇X η)(Z)}. (4.1) b(2n + 1) 2b b Putting Y = φY and Z = φZ in the above equation we get    dr(W ) a + 2b g(φY, φZ). (∇X S)(φY, φZ) = b(2n + 1) 2b (4.2) Thus, we are in a position to state the following: Theorem 4.1. A quasi-conformally flat trans-Sasakian manifold of dimension n ≥ 5 is η-parallel if and only if the scalar curvature is constant, provided that α and β are constants. Again by [8] a trans-Sasakian manifold of dimension n ≥ 5 is either a Cosymplectic manifold or an α-Sasakian manifold or a β-Kenmotsu manifold. Thus we have the following corollaries: Corollary 4.1. A quasi-conformally flat Cosymplectic manifold of dimension n ≥ 5 is η-parallel if and only if the scalar curvature is constant, provided that α and β are constants. LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 35 No. 2 2014 SOME CURVATURE PROPERTIES 61 Corollary 4.2. A quasi-conformally flat α-Sasakian manifold of dimension n ≥ 5 is η-parallel if and only if the scalar curvature is constant, provided that α and β are constants. Corollary 4.3. A quasi-conformally flat β-Kenmotsu manifold of dimension n ≥ 5 is η-parallel if and only if the scalar curvature is constant, provided that α and β are constants. Remark 4.1. In this connection it should be mentioned that in paper [10] the authors have proved that a three dimensional trans-Sasakian manifold is η-parallel if and only if the scalar curvature is constant provided that α and β are constants. Definition 4.2. The Ricci tensor S of a quasi-conformally flat trans-Sasakian manifold will be called cyclic parallel if, (∇X S)(Y, Z) + (∇Y S)(Z, X) + (∇Z S)(X, Y ) = 0, (4.3) for any vector fields X, Y , Z. Using (4.1) we get the following relation,    dr(X) a (∇X S)(Y, Z) + (∇Y S)(Z, X) + (∇Z S)(X, Y ) = + 2b g(Y, Z) b(2n + 1) 2b       a a dr(X) r + − + 2b η(Y )η(Z) + − + 2b b(2n + 1) 2b b(2n + 1) 2b    a + 4n + (α2 − β 2 ) {(∇X η)(Y )η(Z) + η(Y )(∇X η)(Z) b       dr(Y ) a a dr(Y ) + 2b g(Z, X) + − + 2b η(Z)η(X) + b(2n + 1) 2b b(2n + 1) 2b       r a a 2 2 + − + 2b + 4n + (α − β ) {(∇Y η)(Z)η(X) b(2n + 1) 2b b    dr(Z) a + η(Z)(∇Y η)(X) + + 2b g(X, Y ) b(2n + 1) 2b       a a dr(Z) r + − + 2b η(X)η(Y ) + − + 2b b(2n + 1) 2b b(2n + 1) 2b    a 2 2 + 4n + (α − β ) {(∇Z η)(X)η(Y ) + η(X)(∇Z η)(Y ). b (4.4) Taking X, Y and Z orthogonal to ξ we get from above,   a dr(X) + 2b g(Y, Z) (∇X S)(Y, Z) + (∇Y S)(Z, X) + (∇Z S)(X, Y ) = b(2n + 1) 2b       dr(Y ) dr(Z) a a + + 2b g(Z, X) + + 2b g(X, Y ). b(2n + 1) 2b b(2n + 1) 2b  (4.5) Thus, we have the following: Theorem 4.2. A quasi-conformally flat trans-Sasakian manifold of dimension n ≥ 5 is cyclic parallel if and only if the scalar curvature is constant, provided that α and β are constants. Again by [8] a trans-Sasakian manifold of dimension n ≥ 5 is either a Cosymplectic manifold or an α-Sasakian manifold or a β-Kenmotsu manifold. Thus we have the following corollaries: Corollary 4.4. A quasi-conformally flat Cosymplectic manifold of dimension n ≥ 5 is cyclic parallel if and only if the scalar curvature is constant, provided that α and β are constants. Corollary 4.5. A quasi-conformally flat β-Kenmotsu manifold of dimension n ≥ 5 is cyclic parallel if and only if the scalar curvature is constant, provided that α and β are constants. Corollary 4.6. A quasi-conformally flat α-Sasakian manifold of dimension n ≥ 5 is cyclic parallel if and only if the scalar curvature is constant, provided that α and β are constants. Definition 4.3. A quasi-conformally flat trans-Sasakian manifold is called locally φ-Ricci symmetric if (4.6) φ2 (∇W Q)X = 0, LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 35 No. 2 2014 62 ALI AKBAR, AVIJIT SARKAR where the vector fields X and W are orthogonal to ξ. The notion of locally φ-Ricci symmetry was introduced by U.C. De and A. Sarkar [11]. Now we get from (3.16) QX = AX + Bη(X)ξ. Using value of A and B in above we get          a a a r r 2 2 + 2b + 2n − + 2b QX = (α − β ) X + − b(2n + 1) 2b b b(2n + 1) 2b    a 2 2 + 4n + (α − β ) η(X)ξ. b Differentiating covariantly with respect to W we get from above,       d(W )r a a dr(W ) + 2b X + − + 2b (∇W Q)X = b(2n + 1) 2b b(2n + 1) 2b + {(∇W η)(X)ξ + η(X)∇W ξ}. Considering X orthogonal to ξ and applying φ2 on both side of above we get,    dr(W ) a 2 φ (∇W Q)X = − + 2b X. b(2n + 1) 2b (4.7) (4.8) (4.9) (4.10) Thus, we are in a position to state the following: Theorem 4.3. A quasi-conformally flat trans-Sasakian manifold of dimension n ≥ 5 is locally φRicci symmetric if and only if the scalar curvature is constant, provided that α and β are constants. Again by [8] a trans-Sasakian manifold of dimension n ≥ 5 is either a Cosymplectic manifold or an α-Sasakian manifold or a β-Kenmotsu manifold. Thus we have the following corollaries: Corollary 4.7. A quasi-conformally flat Cosymplectic manifold of dimension n ≥ 5 is locally φ-Ricci symmetric if and only if the scalar curvature is constant, provided that α and β are constants. Corollary 4.8. A quasi-conformally flat α-Sasakian manifold of dimension n ≥ 5 is locally φ-Ricci symmetric if and only if the scalar curvature is constant, provided that α and β are constants. Corollary 4.9. A quasi-conformally flat β-Kenmotsu manifold of dimension n ≥ 5 is locally φ-Ricci symmetric if and only if the scalar curvature is constant, provided that α and β are constants. 5. EINSTEIN SEMI-SYMMETRIC QUASI-CONFORMALLY FLAT TRANS-SASAKIAN MANIFOLD Definition 5.1. The Einstein Tensor , denoted by E is defined by r (5.1) E(X, Y ) = S(X, Y ) − g(X, Y ), 2 where S is Ricci tensor and r is scalar curvature. Definition 5.2. A (2n + 1)-dimensional quasi-conformally flat trans-Sasakian manifold will be called Einstein Semi-symmetric if, R(X, Y ).E(Z, W ) = 0 (5.2) R(X, Y )Z = M {g(Y, Z)X − g(X, Z)Y } + N {g(Y, Z)η(X)ξ − g(X, Z)η(Y )ξ + η(Y )η(Z)X − η(X)η(Z)Y }. (5.3) for any vector fields X, Y , Z and W . Now from equation (3.17) we get where, M=       2b a a r 2 2 + 2b − 2n − (α − β ) − a(2n + 1) 2b a b (5.4) LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 35 No. 2 2014 SOME CURVATURE PROPERTIES 63 and N=       r a b a 2 2 + 2b − 4n + (α − β ) . a(2n + 1) 2b a b (5.5) Now, we consider the quasi-conformally flat trans-Sasakian manifold which is Einstein Semisymmetric, i.e. R.E = 0. (5.6) Which implies E(R(X, Y )Z, U ) + E(Z, R(X, Y )U ) = 0. Using (5.1) we get from above r r S(R(X, Y )Z, U ) − g(R(X, Y )Z, U ) + S(Z, R(X, Y )U ) − g(Z, R(X, Y )Z) = 0. 2 2 Using (3.13) we get from above,     r r g(R(X, Y )Z, U ) + A − g(Z, R(X, Y )U ) A− 2 2 + Bη(R(X, Y )Z)η(U ) + Bη(Z)η(R(X, Y )U ) = 0. Putting Z = ξ in above we get,     r r A− g(R(X, Y )ξ, U ) + A − g(ξ, R(X, Y )U ) 2 2 + Bη(R(X, Y )Zξ)η(U ) + Bη(Zξ)η(R(X, Y )U ) = 0. (5.7) (5.8) (5.9) (5.10) Using (5.3) in above we get, B{g(X, U )η(Y ) − g(Y, U )η(X)} = 0. (5.11) B{g(X, U ) − η(U )η(X)} = 0. (5.12) Putting Y = ξ in above we get, Putting U = QW in above and using (3.16) we get, B{g(X, QW ) − (A + B)η(W )η(X)} = 0. or, B{S(X, W ) − (A + B)η(W )η(X)} = 0. This implies that, either B = 0, or S(X, W ) − (A + B)η(W )η(X) = 0. Now if B = 0, then we get from (3.15) that r is constant. Again if, S(X, W ) − (A + B)η(W )η(X) = 0 , then we have S(X, W ) = (A + B)η(W )η(X) (5.13) (5.14) (5.15) Putting X = W = ei in the above equation, where {ei } is an orthonormal basis of the tangent space at any point of the manifold and taking summation over i, 1 ≤ i ≤ 2n + 1, we get r = 6n(α2 − β 2 ) (5.16) Thus, we are in a position to state the following: Theorem 5.1. If a quasi-conformally flat trans-Sasakian manifold of dimension n ≥ 5 is Einstein Semi-symmetric, then the scalar curvature is constant, provided that α and β are constant. Again by [8] a trans-Sasakian manifold of dimension n ≥ 5 is either a Cosymplectic manifold or an α-Sasakian manifold or a β-Kenmotsu manifold. Thus we have the following corollaries: Corollary 5.1. If a quasi-conformally flat Cosymplectic manifold of dimension n ≥ 5 is Einstein Semi-symmetric, then the scalar curvature is constant, provided that α and β are constant. Corollary 5.2. 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